In the production of synthetic yarns, fibers, or fleeces it is generally known that a plurality of fine strand-like filaments is first extruded from a polymer melt. For this, the polymer melt is pressed through fine capillary holes of a spinning device, preferably a spinneret, so that the polymer melt exits from the capillary holes in fine, strand-like filaments. After the extrusion of the filaments it is necessary to cool them to solidify the polymer melt. For this, the filaments are guided through a cooling shaft where, at least from one inner side of the cooling shaft, cool air is blown, via a blower wall, into the cooling shaft. The blower wall, which, for example, can be formed by a wire mesh or several sieves, is disposed at a distance from the filament bundles so that the cool air exiting from the blower wall flows into the cooling shaft essentially in the direction transverse to the filaments and leads to the cooling of the filaments. A process of this type and a device of this type follow, for example, from DE 100 53 731 A1.